|
|
A067874
|
|
Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.
|
|
9
|
|
|
2, 4, 6, 12, 14, 16, 18, 20, 22, 30, 32, 34, 36, 38, 40, 42, 52, 54, 56, 58, 60, 66, 68, 70, 72, 78, 84, 86, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 112, 114, 128, 130, 132, 138, 140, 142, 144, 150, 156, 158, 160, 162, 164, 166, 178, 180, 182, 184, 186, 192, 194, 196, 198
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
D is unique iff x^2 - 1 is squarefree, in which case it follows with necessity that D=x^2-1 and y=1.
All terms are even. A014574 is a subsequence.
The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Feb 25 2021
|
|
LINKS
|
|
|
FORMULA
|
|
|
MAPLE
|
select(t -> numtheory:-issqrfree(t^2-1), [seq(n, n=2..1000, 2)]); # Robert Israel, Apr 12 2016
|
|
MATHEMATICA
|
|
|
PROG
|
(Magma) [n: n in [1..110] | IsSquarefree(n-1) and IsSquarefree(n+1)]; // Juri-Stepan Gerasimov, Jan 17 2017
(Python)
from itertools import count, islice
from sympy import factorint
def A067874_gen(startvalue=2): # generator of terms >= startvalue
return filter(lambda k:max(factorint(k-1).values(), default=1)==1 and max(factorint(k+1).values())==1, count(max(startvalue+(startvalue&1), 2), 2))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nice,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|